The shock theory in multiple dimensions are way more limited than that in one dimension. For the multi-d hyperbolic conservation laws, to our knowledge, the stability of planar inviscid Lax shocks still remains in a linear sense established by [1]}, in which the perturbed shock waves possess piecewise smooth structures. If the solution space of the 2-d compressible Euler equations is relaxed to the \( L^\infty \)  space, [2]}, [3]}, [4]} showed the non-uniqueness of the admissible solutions, in which the initial data are some Riemann data containing at least one planar Lax shock. We also refer to [5]} for the non-uniqueness of spatially periodic admissible solutions for the 2-d compressible Euler equations.
The viscous shock theory for general multi-d viscous conservation laws is also unsatisfactory.
Nevertheless, the nonlinear stability of planar viscous shocks was proved by [6]}, [7]}, [8]} in the scalar case, based on detailed pointwise estimates of Green's functions.
For the multi-d systems, a linear or spectral stability of planar viscous shocks was established by a spectral method; see, for instance, [9]}, [10]} and the references therein.
In particular, [10]} investigated a so-called strong spectral stability condition, which is sufficient for the nonlinear stability of planar viscous shocks.
However, so far the condition can be verified only in a numerical way, and a rigorous mathematical proof is somehow incomplete.
We also refer to [12]}, [13]}, [14]} for some other interesting results about the nonlinear stability of planar viscous shocks.
In particular, the recent paper [14]} showed a nonlinear stability (but not a nonlinear time-asymptotic stability) of planar viscous shocks for the 3-d isentropic N-S equations on the nozzle \( \mathbb {R}\times \mathbb {T}^2, \)  in which the initial perturbation belongs to the \( H^2(\mathbb {R}\times \mathbb {T}^2) \)  space. We give a detailed discussion about [14]} in Rem-loc, especially for the difference between [14]} and this paper about both the analysis and the stability result.
